A finite p-group question: can this happen?A finite 2-group containing the dihedral group of order 16?On the existence of a direct summand containing a fixed subgroupA second isomorphism theorem for the inclusions of groupsIs there a big solvable subgroup in every finite group?Normal subgroups of an extension of the Higman groupPositivity of the alternating sum of indices for boolean interval of finite groupsProper subgroups which have the same minimal subgroups(revision)Minimal SubgroupsFinite groups with cyclic self-normalizing subgroupExistence of a cyclic non-normal subgroup in a $p$-groupCyclic subgroups of finite $p$-groups
A finite p-group question: can this happen?
A finite 2-group containing the dihedral group of order 16?On the existence of a direct summand containing a fixed subgroupA second isomorphism theorem for the inclusions of groupsIs there a big solvable subgroup in every finite group?Normal subgroups of an extension of the Higman groupPositivity of the alternating sum of indices for boolean interval of finite groupsProper subgroups which have the same minimal subgroups(revision)Minimal SubgroupsFinite groups with cyclic self-normalizing subgroupExistence of a cyclic non-normal subgroup in a $p$-groupCyclic subgroups of finite $p$-groups
$begingroup$
Let all groups here be finite $p$--groups.
Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 lhd L_1 lhd cdots lhd L_r = H$, such that each $L_i/L_i-1$ is cyclic.
Question: Let $H$ be a subgroup of $G$. If $K_1$ and $K_2$ are subgroups of $H$ that are conjugate in $G$, does it follow that $r(K_1,H) = r(K_2,H)$?
As in the last question I posed, a minimal counterexample would have $G = langle H,grangle$, and $H$ not normal in $G$.
at.algebraic-topology gr.group-theory finite-groups
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
$endgroup$
add a comment |
$begingroup$
Let all groups here be finite $p$--groups.
Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 lhd L_1 lhd cdots lhd L_r = H$, such that each $L_i/L_i-1$ is cyclic.
Question: Let $H$ be a subgroup of $G$. If $K_1$ and $K_2$ are subgroups of $H$ that are conjugate in $G$, does it follow that $r(K_1,H) = r(K_2,H)$?
As in the last question I posed, a minimal counterexample would have $G = langle H,grangle$, and $H$ not normal in $G$.
at.algebraic-topology gr.group-theory finite-groups
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
$endgroup$
3
$begingroup$
Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question.
$endgroup$
– LSpice
Jun 28 at 0:38
add a comment |
$begingroup$
Let all groups here be finite $p$--groups.
Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 lhd L_1 lhd cdots lhd L_r = H$, such that each $L_i/L_i-1$ is cyclic.
Question: Let $H$ be a subgroup of $G$. If $K_1$ and $K_2$ are subgroups of $H$ that are conjugate in $G$, does it follow that $r(K_1,H) = r(K_2,H)$?
As in the last question I posed, a minimal counterexample would have $G = langle H,grangle$, and $H$ not normal in $G$.
at.algebraic-topology gr.group-theory finite-groups
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
$endgroup$
Let all groups here be finite $p$--groups.
Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 lhd L_1 lhd cdots lhd L_r = H$, such that each $L_i/L_i-1$ is cyclic.
Question: Let $H$ be a subgroup of $G$. If $K_1$ and $K_2$ are subgroups of $H$ that are conjugate in $G$, does it follow that $r(K_1,H) = r(K_2,H)$?
As in the last question I posed, a minimal counterexample would have $G = langle H,grangle$, and $H$ not normal in $G$.
at.algebraic-topology gr.group-theory finite-groups
at.algebraic-topology gr.group-theory finite-groups
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
edited Jun 28 at 0:39
LSpice
3,1182 gold badges26 silver badges31 bronze badges
3,1182 gold badges26 silver badges31 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
asked Jun 28 at 0:12
Nicholas KuhnNicholas Kuhn
4,47113 silver badges27 bronze badges
4,47113 silver badges27 bronze badges
3
$begingroup$
Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question.
$endgroup$
– LSpice
Jun 28 at 0:38
add a comment |
3
$begingroup$
Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question.
$endgroup$
– LSpice
Jun 28 at 0:38
3
3
$begingroup$
Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question.
$endgroup$
– LSpice
Jun 28 at 0:38
$begingroup$
Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question.
$endgroup$
– LSpice
Jun 28 at 0:38
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It does not follow. Let $G=Z_4wr Z_2$, or equivalently $$G=langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=brangle.$$ Let $H=langle a,b^2rangle$, $K_1=langle a^2rangle$, and $K_2=langle b^2rangle$. Then $K_1^t=K_2$, $r(K_1,H)=2$, and $r(K_2,H)=1$.
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
$endgroup$
2
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
add a comment |
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1 Answer
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$begingroup$
It does not follow. Let $G=Z_4wr Z_2$, or equivalently $$G=langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=brangle.$$ Let $H=langle a,b^2rangle$, $K_1=langle a^2rangle$, and $K_2=langle b^2rangle$. Then $K_1^t=K_2$, $r(K_1,H)=2$, and $r(K_2,H)=1$.
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
$endgroup$
2
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
add a comment |
$begingroup$
It does not follow. Let $G=Z_4wr Z_2$, or equivalently $$G=langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=brangle.$$ Let $H=langle a,b^2rangle$, $K_1=langle a^2rangle$, and $K_2=langle b^2rangle$. Then $K_1^t=K_2$, $r(K_1,H)=2$, and $r(K_2,H)=1$.
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
$endgroup$
2
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
add a comment |
$begingroup$
It does not follow. Let $G=Z_4wr Z_2$, or equivalently $$G=langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=brangle.$$ Let $H=langle a,b^2rangle$, $K_1=langle a^2rangle$, and $K_2=langle b^2rangle$. Then $K_1^t=K_2$, $r(K_1,H)=2$, and $r(K_2,H)=1$.
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
$endgroup$
It does not follow. Let $G=Z_4wr Z_2$, or equivalently $$G=langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=brangle.$$ Let $H=langle a,b^2rangle$, $K_1=langle a^2rangle$, and $K_2=langle b^2rangle$. Then $K_1^t=K_2$, $r(K_1,H)=2$, and $r(K_2,H)=1$.
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
answered Jun 28 at 0:54
Richard LyonsRichard Lyons
1,0248 silver badges10 bronze badges
1,0248 silver badges10 bronze badges
2
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
add a comment |
2
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
2
2
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
$begingroup$
Thanks! (Oh man, I should have thought of that.)
$endgroup$
– Nicholas Kuhn
Jun 28 at 1:02
add a comment |
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$begingroup$
Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question.
$endgroup$
– LSpice
Jun 28 at 0:38